TFPT scientific source excerpt Original repository path: experiments/theory-contracts/native-octic-seam-origin-20261025/P1-SOURCE-EMBEDDING-AUDIT.txt Original file SHA-256: daba8a2cc5ea350c1e2b3d9b005b8fe974a7b67590f3c0a32ec5b0d053ec8f6e Inclusive original lines: 1-195 Line numbers below refer to the original source. This bounded excerpt is source evidence; it is not a new proof/checker run. Conversation metadata and personal execution details are omitted where indicated. 000001 | P1 SOURCE EMBEDDING AUDIT 000002 | ========================= 000003 | 000004 | 25 October 2026. Read only source audit followed by this scoped report. 000005 | Verdict: PARTIAL EXACT SOURCE EMBEDDING. No ledger or physical status promotion. 000006 | 000007 | 1. Decisive result 000008 | ------------------ 000009 | 000010 | The supplied quarter CAR fold already contains the field, state, Euclidean 000011 | time and nontrivial seam needed for the native octic response. The response 000012 | does not require a charged E8 root or spinor extension. It is an even composite 000013 | of the eight original CAR channels. 000014 | 000015 | This does not yet prove that the supplied source is an object of Seedop. The 000016 | archived definition at 000017 | 000018 | documents/archive--tfpt-45--source_extracts--01_boundary_kernel_source/ 000019 | sections/002-section-intrinsic-closed-branch-and-canonical-rigid-obje.tex: 000020 | 103--143 000021 | 000022 | requires one tuple 000023 | 000024 | (A_loc,tau_t,Theta,omega,[u_Sigma],D_coll) 000025 | 000026 | with reflection positivity, clustering, finite relative action against its 000027 | declared reference, and a reflection regular elliptic collar. The existing 000028 | CAR construction supplies a canonical bounded even net, state and Euclidean 000029 | semigroup, but no source proves that its reflection, a collar lift and the 000030 | archived reference are the three displayed P1 objects. Operational collar 000031 | completion runs from a complete Seedop object to boundary data. It is not an 000032 | inverse theorem constructing these missing entries. 000033 | 000034 | The exact advance is therefore substantial and narrow. The former general 000035 | field identification problem has reduced to one common intertwiner test for 000036 | Theta, D_coll and D_ref. 000037 | 000038 | 2. The octic is already an original even CAR field 000039 | -------------------------------------------------- 000040 | 000041 | Let 000042 | 000043 | j_r(z)=:chi_r^*(z) chi_r(z):_sea, r=0,...,7, 000044 | A=U_+^T O^T M, b=A delta. 000045 | 000046 | Here chi denotes the original fold eigenchannels, M is the original Weyl 000047 | map after folding, O is the existing E8 current chart, and U_+ is the fixed 000048 | J positive polarization. Direct exact multiplication gives 000049 | 000050 | A A^dagger=I_4, A A^T=0. 000051 | 000052 | Thus A selects four mutually isotropic linear combinations of the same eight 000053 | source currents. With the exact octic I8 and the original quarter shift delta, 000054 | 000055 | Y_d(F8,z) 000056 | = : I8(A(j(z)+delta/z))/(96 sqrt(140)) : 000057 | = : I8(h_+(z)+b/z)/(96 sqrt(140)) : . (1) 000058 | 000059 | Equation (1) is a neutral even CAR composite on the same eight preimage arcs. 000060 | The normal ordering is the retained bosonic current normal order, equivalently 000061 | the original point split CAR limit with all finite derivative counterterms. 000062 | It is not the naive fully fermion normal ordered degree sixteen monomial. 000063 | 000064 | The precise source prescription is already recorded in 000065 | 000066 | experiments/theory-contracts/joint-native-algebra-selection-20261016/ 000067 | ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt:108--160. 000068 | 000069 | Those lines give the arc fields, the bilinear matrices, the unchanged 000070 | subtraction, the single b shift, and every separated even CAR correlation by 000071 | the original quasifree Pfaffian. Lines 64--68 distinguish this field map from 000072 | the Hall state/current isomorphism. The affine current and time dictionary is 000073 | also explicit in 000074 | 000075 | experiments/theory-contracts/joint-native-algebra-selection-20261016/ 000076 | FOLD-COVARIANCE-TO-NATIVE-FIELD-RESPONSE.txt:53--115,117--149. 000077 | 000078 | The physical source algebra is the full original gauge even CAR observable 000079 | net. No freely selected Heisenberg sector is being substituted for it. The 000080 | neutral currents and (1) are affiliated subfields of that already fixed net. 000081 | 000082 | 3. Locality statement that the sources actually prove 000083 | ------------------------------------------------------ 000084 | 000085 | The original net remains the point local c=1 CAR net. A quotient interval is 000086 | the union of eight separated preimage arcs, and the transported state retains 000087 | their correlations. In the quotient localization the eight component stress 000088 | has c=8. These are two specified localization structures on the same fold, 000089 | not an equality of point local c=1 and c=8 nets. The exact source is 000090 | 000091 | experiments/theory-contracts/v16-operator-pullback-20261014/ 000092 | FOLD-AUDIT.md:10--65,67--86. 000093 | 000094 | Consequently (1) is quasilocal in the original source on the union of the 000095 | eight arcs and local in the stipulated quotient chart. This is enough to place 000096 | the response inside the already given CAR source. It does not prove that P1 000097 | physically selects the quotient chart as its point localization. 000098 | 000099 | 4. Reflection positivity and clustering on the bounded even source algebra 000100 | --------------------------------------------------------------------------- 000101 | 000102 | The supplied sea relative CAR representation has the occupation basis of 000103 | finite particle and hole sets and the positive selfadjoint excitation 000104 | Hamiltonian 000105 | 000106 | H|S>=E(S)|S>, 000107 | E(S)=sum_(sigma,n in S) epsilon_(sigma,n), 000108 | epsilon_(+,n)=n+alpha, 000109 | epsilon_(-,n)=n+1-alpha. 000110 | 000111 | For alpha=1/4 or 3/4 the vacuum is unique and the original cover generator has 000112 | gap 1/4. The quotient generator H/8 has gap 1/32. The factor eight is part of 000113 | the fold and is not a new time calibration. The construction and domains are 000114 | given in 000115 | 000116 | experiments/theory-contracts/seam-collar-multifield-20261016/ 000117 | PROOF.txt:199--226. 000118 | 000119 | For a bounded even history at 0<=t_1<=...<=t_n define 000120 | 000121 | v_(A,t)=exp(-t_1 H) A_1 exp(-(t_2-t_1)H) A_2 ... 000122 | exp(-(t_n-t_(n-1))H) A_n Omega. 000123 | 000124 | For a finite linear combination F of such histories put 000125 | v_F=sum_j c_j v_(A_j,t_j). This includes arbitrary numbers of insertions; 000126 | all intervening semigroups are bounded contractions. 000127 | 000128 | The source adjoint and its same Euclidean semigroup give the canonical history 000129 | reflection, and its OS form is 000130 | 000131 | OS(F,F)=||v_F||^2 >=0. (2) 000132 | 000133 | No assumption that an arbitrary field acts by left multiplication on an OS 000134 | quotient is used in (2). It is a direct Hilbert space formula for bounded even 000135 | histories in the supplied representation. Unbounded current and octic fields 000136 | are handled on their common finite energy and point split domains; (2) is the 000137 | bounded net statement. 000138 | 000139 | The same gap proves Euclidean imaginary time clustering. If P0 is the vacuum 000140 | projection and A,B are bounded even local operators, then 000141 | 000142 | C_AB(s)=, 000143 | |C_AB(s)| <= exp(-s Delta) 000144 | ||(1-P0)A Omega|| ||(1-P0)B Omega||, (3) 000145 | 000146 | with Delta=1/4 for the cover generator or Delta=1/32 for its fixed quotient 000147 | normalization. Equation (3) makes no assertion that real circle time 000148 | correlations decay. It also does not identify this canonical CAR reflection 000149 | with the geometrically declared P1 reflection Theta. 000150 | 000151 | The general Osterwalder Schrader reconstruction theorem applies when the full 000152 | Euclidean Schwinger family and all OS axioms are supplied. The present source 000153 | does supply its quasifree CAR family and finite even composites, but the theorem 000154 | does not manufacture the P1 collar or the declared comparison reference. 000155 | Primary reference: Osterwalder and Schrader, Communications in Mathematical 000156 | Physics 42 (1975), 281--305, 000157 | https://projecteuclid.org/journals/communications-in-mathematical-physics/ 000158 | volume-42/issue-3/Axioms-for-Euclidean-Greens-functions-II-with-an-Appendix-by/ 000159 | cmp/1103899050.pdf 000160 | 000161 | 5. The seam class is already nontrivial 000162 | --------------------------------------- 000163 | 000164 | For the actual circle operator and seam multiplier 000165 | 000166 | B_alpha=-i partial_theta+alpha, Z=exp(i theta), 000167 | 000168 | one has [B_alpha,Z]=Z. The path B_s=B_alpha+s has exactly one upward crossing, 000169 | so its spectral flow is +1. Equivalently Z has winding one. This gives the 000170 | nontrivial source seam class without a fitted profile. The complete scalar 000171 | statement and its scope are in 000172 | 000173 | experiments/theory-contracts/joint-native-algebra-selection-20261016/ 000174 | SCALAR-FIRST-ORDER-SOURCE-FIBER-AND-SEAM-WARD.txt:67--105. 000175 | 000176 | The charged seam action on all current products, including its distinction 000177 | from quarter holonomy, is retained in 000178 | 000179 | experiments/theory-contracts/joint-native-algebra-selection-20261016/ 000180 | ORIGINAL-ORDERED-CURRENT-KERNEL-AND-SEAM-GRADING.txt:162--181. 000181 | 000182 | 6. The optional full E8 extension is not needed 000183 | ----------------------------------------------- 000184 | 000185 | In the shifted full E8 module the root beta=e1+e4 obeys d.beta=-1, so its 000186 | shifted weight is 1+d.beta=0. This observation cannot be used against the 000187 | actual CAR source. Exact pullback through the original fold gives 000188 | 000189 | M^T beta=(1/2,1/2,1/2,1/2,-1/2,-1/2,-1/2,-1/2), (4) 000190 | 000191 | which is half integral and is not an original CAR charge in Z^8. The neutral 000192 | current polynomial (1) therefore exists before this root is adjoined. 000193 | 000194 | If one optionally enlarges the algebra to the full E8 charge extension while 000195 | keeping Omega_d unchanged, the extra zero energy channel means that the CAR